Image Processing Reference
In-Depth Information
1.8 Composition of Intuitionistic Fuzzy Relation
Using Fuzzy
t
-Norm and
t
-Conorm
Composition of IFR can also be carried out using
t
-norms and
t
-conorms
[2- 4].
T
-norms and
t
-conorms are a kind of binary operation used to rep-
resent the intersection and union in fuzzy set theory, respectively. In this
section, fuzzy
t
-norm and
t
-conorm are discussed briefly, and these are dis-
cussed in detail in Chapter 3.
1.8.1
t
-Norm
A
t
-norm
T
: [0, 1] → [0, 1] is a kind of binary operation used in the framework
of fuzzy logic and probabilistic metric spaces. It represents the intersection
in fuzzy set theory or an 'ANDing' operator.
The four basic
t
-norms are
1. The minimum:
T
(
x
,
y
) = min(
x
,
y
)
2. The product:
T
(
x
,
y
) =
x
·
y
3. The Lukasiewicz
t
-norm:
T
L
(
x
,
y
) = max(
x
+
y −
1, 0)
4. The Nilpotent minimum
t
-norm:
Txy
(
,
)
=
min
(
x y
,
)
if
x
+ >
y
1
=
0
otherwise
1.8.2
t
-Conorm
A
t
-conorm
T
: [0, 1] → [0, 1] is a kind of binary operation used in the frame-
work of fuzzy logic and probabilistic metric spaces. It represents union in
fuzzy set theory or an 'ORing' operator. The four basic
t
-conorms are
1. The maximum:
S
(
x
,
y
) = max(
x
,
y
)
2. The product:
S
(
x
,
y
) =
x
+
y
−
x
·
y
3. The Lukasiewicz
t
-conorm:
T
L
(
x
,
y
) = min(
x
+
y
, 1)
4. Nilpotent minimum
t
-conorm:
Sxy
(
,
)
=
max
(
x y
,
)
if
x
+ <
y
1
=
1
otherwise
For IFRs,
t
-norms and
t
-conorms will be designated with letters α, β, ρ and δ
in this chapter.
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